MATHEMATICAL AND COMPUTATIONAL TOOLS (Qing Nie, Theme Leader) The processes and interactions dealt with in Themes A-C are all spatiotemporally dynamic, typically multiscale, and potentially subject to large stochastic effects. Quantitative mathematical and computational analysis of such systems faces substantial challenges, at least using conventional methods. For example, the efficient exploration of large parameter spaces--necessary for model exploration--is hindered by deficiencies in methods for fast, accurate simulation. In Aim Dia, we propose to develop new fast methods for steady state continuum models that involve multiple spatial scales; In Aim Dib, we propose a convenient and robust computational framework with a new efficient algorithm for solving systems involving temporally evolving spatial domains - a type of continuum model especially relevant to tissue growth (e.g. in Theme B) Spatiotemporal stochastic effects pose special challenges. While non-spatial stochastic modeling and simulation has provided many recent insights into biochemical reactions, spatial stochastic methods need much further development. In Aim D2a, we propose a new hybrid spatial model and algorithm that couples continuum stochastic partial differential equations with discrete stochastic reaction-diffusion processes; In Aim D2b, we propose a multi-scale hybrid model and algorithm that accounts for individual cells, continuum descriptions of morphogens, intracellular regulatory networks, and possible mechanical effects. The tools developed in Aim D2a can be applied to the hybrid approach in Aim D2b. These modeling frameworks will help projects in Themes A-C explore stochastic effects more freely and efficiently than is currently possible. A common goal in Systems Biology is to use large biological data sets to learn the topology and parameters of biological networks. Defining complex gene regulatory networks is particularly important for understanding systems that drive spatial phenomena, such as patterning and morphogenesis. Yet, currently, most network inference is done using perturbation-series, or time-series data, but not continuous spatial information. We propose to begin to address this deficiency by starting to develop, in Aim D3, methods for inferring spatiotemporal models from spatiotemporal data. This approach begins with the development of a regularization framework to enable incorporation of different kinds of data into inference algorithms, and continues with development of approaches to use imaging data in network inference. One of our major goals in the development of computational tools is robustness. To meet the need for large scale model exploration that the kinds of biology in this proposal require, we must create methods that workwell over large ranges of parameter space, initial and/or boundary conditions, and model architecture. Although we can always expect trade-offs between computafional robustness and speed, computational frameworks that require minimal fine-tuning to the specifics of individual models are likely to be much more useful to the work in this proposal, and to the Systems Biology community in general.